Mission accomplished
So, trying one more new (used) battery arrangement means the mower now has enough juice to do the entire lawn (~20-25 mins).
I now have one 12 amp-hour battery (which seems to have about 2/3 of its rated capacity - delivered ~4 amps for 120 minutes in my inverter test) in parallel with a 7 amp-hour battery that has ~60% of its rated capacity).
They seem to work well together.
At this point, all that's left to do is to tape up a few connections, secure the batteries, and put the "hood" back on. I may lead a pair of wires through the hood so I can easily hook up a multimeter to monitor voltage and keep an eye on that battery pack.
It's safe to say this free mower is back in service with free used batteries (with some time invested testing their capacities). Alternatively, about $120 would have bought a new battery that would have had it back on the road (um, lawn) with little muss or fuss. (But where's the fun in that?!)
MISSION ACCOMPLISHED!